Strange Attractor

The stroboscopic surface of section we produced in the previous section is a Strange Attractor.

It is an attractor in the sense that trajectories originating from various initial conditions converge to it. It is Strange in the sense that it is neither a continuous line in $x$ - $v$ phase space (like a Limit Cycle which forms a continuous trajectory in phase space) nor is it filling the $x$ - $v$ plane. It is a Fractal.

As in the case of the Driven One Well chaotic solution, this solution exhibits an extreme sensitivity to initial conditions. This is the hallmark of chaos.

This sensitivity to initial conditions and the attractiveness of the Strange Attractor can be illustrated by following a bunch of trajectories based on initial conditions that are fairly close together. These trajectories should all converge to the the Strange Attractor if observed in a stroboscopic map at $t=nT$. Therefore they will build the shape of the Strange Attractor point by point. These points will be far apart on the Strange Attractor due to the exponential divergence of the orbits. This Strange Attractor is really Strange!

Exercise #31:

Write a 2 wells chaos initial setup code to generate initial conditions to launch trajectories that scan a grid $N_x \times N_v$ on a small rectangle in $x$-$v$ space. Use $N_x = 50$, $N_v = 50$ spanning $x_0 = [-1.1,-1.0]$ and $v_0 = [-0.1,0.1]$ to start with. These initial conditions will be fed into the ODE solving code. Use $\alpha = 1.0$, $\beta = -1$, $\mu = 0.25$, $A = 0.4$ and $\omega = 1$ in the code implementing the RK4 solution of Duffing Oscillator. Make sure that the code outputs the last $x$ and $v$ values when the trajectory data are not printed out ( by not using the -p option ).

Run the code for 1, 2, 3, 4, 5, 10, 20, 100 outside driving term periods ( via the option -t n ) thereby producing stroboscopic maps data to be plotted separately. This will illustrates divergence of the trajectories and the attraction of the Strange Attractor.

Exercise #32:

Repeat the step above with more initial conditions, e.g., 20000, to refine the image of the Strange Attractor.

Exercise #33:

Start from a different region of phase space.